Notes on the LVP and CVP in p-Adic Fields

This paper explores computational methods for solving the Longest Vector Problem (LVP) and Closest Vector Problem (CVP) in p -adic fields. Leveraging the non-Archimedean property of p -adic norms, we propose a polynomial time algorithm to compute orthogonal bases for p -adic lattices when the p -adic field is given by a minimal polynomial. The method utilizes the structure of maximal orders and p -radicals in extension fields of Qₚ to efficiently construct uniformizers and residue field bases, enabling rapid solutions for the LVP and CVP. In addition, we introduce the characterization of norms on vector spaces over Qₚ .